A few families of non-schurian association schemes 1

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1 A few families of non-schurian association schemes 1 Štefan Gyürki Slovak University of Technology in Bratislava, Slovakia Ben-Gurion University of the Negev, Beer Sheva, Israel CSD6, Portorož Joint work with M. Klin Štefan Gyürki A few families of non-schurian association schemes 1/ 40

2 Preliminaries Definitions Computer facilities Non-Schurian association schemes on 18 points Color graph Under a color graph Γ we will mean an ordered pair (V, R), where V is a set of vertices and R a partition of V V into binary relations. The elements of R will be called as colors, and the number of colors is the rank of Γ. Štefan Gyürki A few families of non-schurian association schemes 2/ 40

3 Preliminaries Definitions Computer facilities Non-Schurian association schemes on 18 points Coherent configuration A coherent configuration is a color graph N = (Ω, R), R = {R i i I }, such that the following axioms are satisfied: (i) The diagonal relation Ω = {(x, x) x Ω} is a union of relations i I R i, for a suitable subset I I. (ii) For each i I there exists i I such that Ri T = R i, where Ri T = {(y, x) (x, y) R i } is the relation transposed to R i. (iii) For any i, j, k I, the number c k i,j of elements z Ω such that (x, z) R i and (z, y) R j is a constant depending only on i, j, k, and independent on the choice of (x, y) R k. Štefan Gyürki A few families of non-schurian association schemes 3/ 40

4 Preliminaries Definitions Computer facilities Non-Schurian association schemes on 18 points The numbers c k i,j are called intersection numbers, or sometimes structure constants of N. An association scheme N = (Ω, R) is a homogeneous coherent configuration, i.e. where the diagonal relation Ω does belong to R. A coherent configuration N is called commutative, if for all i, j, k I we have c k ij = c k ji ; and it is called symmetric if R i = R T i for all i I. Štefan Gyürki A few families of non-schurian association schemes 4/ 40

5 Preliminaries Definitions Computer facilities Non-Schurian association schemes on 18 points The orbits of a group G on the set Ω Ω are called 2-orbits, or orbitals. If 2 Orb(G, Ω) is the set of 2-orbits of a permutation group (G, Ω), then (Ω, 2 Orb(Ω)) is a coherent configuration. Those coherent configurations which can be obtained in this manner are called Schurian, otherwise non-schurian. Thus, Schurian association schemes are coming from transitive permutation groups. Štefan Gyürki A few families of non-schurian association schemes 5/ 40

6 Preliminaries Definitions Computer facilities Non-Schurian association schemes on 18 points The (combinatorial) group of automorphisms Aut(N ) consists of permutations φ : Ω Ω which preserve the relations, i.e. R φ i = R i for all R i R. The color automorphisms preserve relations setwise, i.e. for φ : Ω Ω we have φ CAut(N ) if and only if for all i I there exists j I such that R φ i = R j. An algebraic automorphism is a bijection φ : R R which satisfies c k ij = c kφ i φ j φ. Štefan Gyürki A few families of non-schurian association schemes 6/ 40

7 Preliminaries Definitions Computer facilities Non-Schurian association schemes on 18 points Let G be a subgroup of the group of algebraic automorphisms of a coherent configuration. Let R/G denote the set of orbits of G on R. For each O R/G define O + to be the union of all relations from O. Then the set of relations {O + O R/G} forms a coherent configuration on Ω. We will call it as algebraic merging of R with respect to G. Štefan Gyürki A few families of non-schurian association schemes 7/ 40

8 Definitions Computer facilities Non-Schurian association schemes on 18 points Why to study association schemes? Applications codes Algebra designs statistical questions. It is a nice table algebra. Transitive group actions on finite sets, distance-regular graphs, finite buildings can be viewed as association schemes. Štefan Gyürki A few families of non-schurian association schemes 8/ 40

9 Computer facilities Definitions Computer facilities Non-Schurian association schemes on 18 points The main results are obtained as theoretical generalizations of observations, which were earned with the aid of a computer. We used the computer algebra system GAP, in conjunction with GRAPE and nauty, packages COCO (Faradžev-Klin, 1991), COCO II (Reichard) and a package of elementary functions for association schemes on GAP (Hanaki, Miyamoto). Štefan Gyürki A few families of non-schurian association schemes 9/ 40

10 Computer facilities Definitions Computer facilities Non-Schurian association schemes on 18 points COCO was the first computer package for computing with coherent configurations developped in 1991 in Moscow by Faradžev s team. induced action of a permutation group on a combinatorial structure; the centralizer algebra of a permutation group; the intersection numbers; to find fusions; to calculate the (combinatorial) automorphism group. Štefan Gyürki A few families of non-schurian association schemes 10/ 40

11 Computer facilities Definitions Computer facilities Non-Schurian association schemes on 18 points COCO II (S. Reichard) Works under GAP. New functions (color automorphisms, algebraic automorphisms,... ). Still under construction. Štefan Gyürki A few families of non-schurian association schemes 11/ 40

12 Definitions Computer facilities Non-Schurian association schemes on 18 points Computer facilities Webpage of Hanaki and Miyamoto Classification of association schemes with small number of vertices (< 39, but not 31, 35, 36, 37). Elementary functions for association schemes on GAP. Štefan Gyürki A few families of non-schurian association schemes 12/ 40

13 Definitions Computer facilities Non-Schurian association schemes on 18 points Non-Schurian association schemes on 18 points A careful analysis of known association schemes on 18 points, which are available from the homepage of Hanaki and Miyamoto, was our starting point. The main interest was to understand two non-schurian association schemes. Štefan Gyürki A few families of non-schurian association schemes 13/ 40

14 Definitions Computer facilities Non-Schurian association schemes on 18 points Non-Schurian association schemes on 18 points The color matrix of the non-schurian association scheme on 18 points of rank 8 (nr. 62 in the catalogue). Štefan Gyürki A few families of non-schurian association schemes 14/ 40

15 Definitions Computer facilities Non-Schurian association schemes on 18 points Non-Schurian association schemes on 18 points The color matrix of the non-schurian association scheme on 18 points of rank 6 (nr. 41 in the catalogue). Štefan Gyürki A few families of non-schurian association schemes 15/ 40

16 What was done? Preliminaries Definitions Computer facilities Non-Schurian association schemes on 18 points Finally, we realized that, for each prime p, we can: work with an intransitive permutation group G of order p 3, acting on two orbits of length p 2, construct a corresponding coherent configuration M of rank 6p 2 with two fibers, detect in M four association schemes. Štefan Gyürki A few families of non-schurian association schemes 16/ 40

17 Biaffine planes Preliminaries Biaffine planes The biaffine coherent configuration Four color graphs Groups of combinatorial automorphisms The second model of M The biaffine plane B p consists of two copies of Z p Z p : points P and non-vertical lines L. Points: P = [x, y]. Lines: l = (k, q), y = k x + q. Incidence: P = [x, y] is incident to l = (k, q) if and only if y = k x + q. Štefan Gyürki A few families of non-schurian association schemes 17/ 40

18 Biaffine planes Preliminaries Biaffine planes The biaffine coherent configuration Four color graphs Groups of combinatorial automorphisms The second model of M P oints : Lines l k = 0 : Lines l k = 1 : Lines l k = 2 : [0, 2] [1, 2] [2, 2] [0, 1] [1, 1] [2, 1] [0, 0] [1, 0] [2, 0] Figure: The objects of the biaffine plane B 3. Štefan Gyürki A few families of non-schurian association schemes 18/ 40

19 The biaffine coherent configuration Biaffine planes The biaffine coherent configuration Four color graphs Groups of combinatorial automorphisms The second model of M Take an action of the permutation group G = (Z p ) 2 Z p on the set Ω = P L. At this stage it appears as deus ex machina. Štefan Gyürki A few families of non-schurian association schemes 19/ 40

20 The biaffine coherent configuration Biaffine planes The biaffine coherent configuration Four color graphs Groups of combinatorial automorphisms The second model of M G = t 1,0, t 0,1, φ, where t a,b : [x, y] [x + a, y + b], φ : [x, y] [x, y x], (k, q) (k, b + q ak), (k, q) (k 1, q). Štefan Gyürki A few families of non-schurian association schemes 20/ 40

21 The biaffine coherent configuration Biaffine planes The biaffine coherent configuration Four color graphs Groups of combinatorial automorphisms The second model of M The group G has 6p 2 orbits on Ω Ω: (P 1, P 2 ) A i x 1 = x 2 and y 2 y 1 = i, where i Z p, (P 1, P 2 ) B i x 2 x 1 = i 0, where i Z p \ {0}, (l 1, l 2 ) C i k 1 = k 2 and q 2 q 1 = i, where i Z p, (l 1, l 2 ) D i k 2 k 1 = i 0, where i Z p \ {0}, (P 1, l 1 ) E i k 1 x 1 + q 1 y 1 = i, where i Z p, (l 1, P 1 ) F i y 1 k 1 x 1 q 1 = i, where i Z p. Štefan Gyürki A few families of non-schurian association schemes 21/ 40

22 The biaffine coherent configuration Biaffine planes The biaffine coherent configuration Four color graphs Groups of combinatorial automorphisms The second model of M Definition The structure M = (Ω, 2 Orb(G)) is called as a biaffine coherent configuration. Štefan Gyürki A few families of non-schurian association schemes 22/ 40

23 Four color graphs Preliminaries Biaffine planes The biaffine coherent configuration Four color graphs Groups of combinatorial automorphisms The second model of M R 0 = A 0 C 0, S i = A i C i, where i = 1, 2,..., p 1, T i = B i D i, where i = 1, 2,..., p 1, U i = E i F i, where i = 0, 1, 2,..., p 1. S i = S i S p i, T i = T i T p i, U i = U i U p i. S = S 1 S 2... S p 1, U = U 1 U 2... U p 1. Štefan Gyürki A few families of non-schurian association schemes 23/ 40

24 Four color graphs Preliminaries Biaffine planes The biaffine coherent configuration Four color graphs Groups of combinatorial automorphisms The second model of M Color graph M 1 colors: R 0, S 1,..., S p 1, T 1,..., T p 1, U 0, U 1,..., U p 1 Color graph M 2 colors: R 0, S 1, S 2,..., S (p 1)/2, T 1, T 2,..., T p 1, U 0, U 1, U 2,..., U (p 1)/2 Color graph M 3 colors: R 0, S, T 1, T 2,..., T p 1, U 0, U Color graph M 4 : colors: R 0, S, T 1, T 2,..., T (p 1)/2, U 0, U. Štefan Gyürki A few families of non-schurian association schemes 24/ 40

25 Theorem 1 Preliminaries Biaffine planes The biaffine coherent configuration Four color graphs Groups of combinatorial automorphisms The second model of M Theorem 1 The following holds: (a) M 1, M 2, M 3, M 4 are association schemes. (b) Combinatorial groups of automorphisms of M 1, M 2, M 3, M 4 contain a subgroup isomorphic to G = Z 2 p Z p. Štefan Gyürki A few families of non-schurian association schemes 25/ 40

26 Biaffine planes The biaffine coherent configuration Four color graphs Groups of combinatorial automorphisms The second model of M Groups of combinatorial automorphisms Theorem 2 Let Aut(M 1 ), Aut(M 2 ), Aut(M 3 ) and Aut(M 4 ) are the combinatorial groups of automorphisms of M 1, M 2, M 3 and M 4, respectively. Then the followings hold: (a) Aut(M 1 ) Aut(M 2 ) = Aut(M 3 ) Aut(M 4 ), (b) Aut(M 1 ) = p 3, (c) Aut(M 2 ) = 2p 3, (d) Aut(M 3 ) = 2p 3, (e) Aut(M 4 ) = 8p 3. Štefan Gyürki A few families of non-schurian association schemes 26/ 40

27 Corollary Preliminaries Biaffine planes The biaffine coherent configuration Four color graphs Groups of combinatorial automorphisms The second model of M Corollary 3 For each p > 3 there exist at least four non-schurian association schemes M 1, M 2, M 3, and M 4 with ranks 3p 1, 2p, p + 3, and (p + 7)/2, respectively. Štefan Gyürki A few families of non-schurian association schemes 27/ 40

28 The second model of M Biaffine planes The biaffine coherent configuration Four color graphs Groups of combinatorial automorphisms The second model of M Let V 1 = {(1, x 1, x 2 ) x 1, x 2 Z p }, x 1 V 2 = x 2 x 1, x 2 Z p. 1 Scalar product (1, x 1, x 2 ) y 1 y 2 1 = y 1 + x 1 y 2 x 2. Štefan Gyürki A few families of non-schurian association schemes 28/ 40

29 The second model of M Biaffine planes The biaffine coherent configuration Four color graphs Groups of combinatorial automorphisms The second model of M Let G = g abc = 1 a b + ac 0 1 c a, b, c Z p. Matrix g abc is invertible, and g 1 abc = 1 a b 0 1 c Štefan Gyürki A few families of non-schurian association schemes 29/ 40

30 The second model of M Biaffine planes The biaffine coherent configuration Four color graphs Groups of combinatorial automorphisms The second model of M Proposition 4 The set G together with the operation of matrix-multiplication form a group, which is isomorphic to (Z p ) 2 Z p, and, in fact, it is the Sylow subgroup of SL(3, p). Define{ an action of G on Ω = V 1 V 2 by: x g x g if x V 1 = g 1 x if x V 2, for all g G. x g y g = (x g) (g 1 y) = x gg 1 y = x y. Štefan Gyürki A few families of non-schurian association schemes 30/ 40

31 The second model of M Biaffine planes The biaffine coherent configuration Four color graphs Groups of combinatorial automorphisms The second model of M Proposition 5 Groups G and G are isomorphic. Corollary 6 The scalar product defined in the second model is invariant with respect to G. Thus, all association schemes may be redefined in these new terms. Štefan Gyürki A few families of non-schurian association schemes 31/ 40

32 Some observations and recent proofs Biaffine planes The biaffine coherent configuration Four color graphs Groups of combinatorial automorphisms The second model of M Observation 7 Association schemes M 1, M 2, M 3, M 4 are algebraic mergings of M. Observation 8 AAut(M) = (Z 2 p 1 Z 2 ) AGL(1, p). Theorem 9 AAut(M 1 ) = Z 2 p 1. Štefan Gyürki A few families of non-schurian association schemes 32/ 40

33 The Pappus graph McKay-Miller-Širáň graphs Wenger graphs The graph defined by the relation U 0 for p = 3 is the Pappus graph. Štefan Gyürki A few families of non-schurian association schemes 33/ 40

34 The Pappus graph McKay-Miller-Širáň graphs Wenger graphs McKay-Miller-Širáň graphs Let p be an odd prime and put V p = Z 2 Z p Z p as vertex set of H p. Let ω be a primitive element. If p = 4r + 1 then define X = {1, ω 2, ω 4,..., ω p 3 }, X = {ω, ω 3,..., ω p 2 }. If p = 4r + 3 then define X = {±1, ±ω 2,..., ±ω 2r }, X = {±ω, ±ω 3,..., ±ω 2r+1 }. Štefan Gyürki A few families of non-schurian association schemes 34/ 40

35 The Pappus graph McKay-Miller-Širáň graphs Wenger graphs The adjacency in the graph H p is defined as follows: (0, x, y) is adjacent to (0, x, y ) if and only if y y X, (1, k, q) is adjacent to (1, k, q ) if and only if q q X, (0, x, y) is adjacent to (1, k, q) if and only if y = kx + q. H p = E 0 F 0 A i i X j X C j. H 5 is the well-known Hoffman-Singleton graph. Štefan Gyürki A few families of non-schurian association schemes 35/ 40

36 The Pappus graph McKay-Miller-Širáň graphs Wenger graphs P 0 P 1 P 2 P 3 P Q 0 Q 1 Q 2 Q 3 Q Adjacencies are between i in P j and i jk in Q k for all 0 i, j, k 4. (Robertson) Štefan Gyürki A few families of non-schurian association schemes 36/ 40

37 The Pappus graph McKay-Miller-Širáň graphs Wenger graphs Wenger graphs The graph W n (q) has as vertex set two copies P and L of the (n + 1)-dimensional vector space over F q. The adjacency between points P = [p 1,..., p n+1 ] and lines L = (l 1,..., l n+1 ) is given by the system: l 2 + p 2 = p 1 l 1 l 3 + p 3 = p 1 l 2 l n+1 + p n+1 = p 1 l n.. Štefan Gyürki A few families of non-schurian association schemes 37/ 40

38 The Pappus graph McKay-Miller-Širáň graphs Wenger graphs n = 1 Wenger graphs W 1 (p) are isomorphic to the graphs defined by U 0. For example, W 1 (3) is isomorphic to the Pappus graph. Wenger graphs belong to a richer family of graphs defined by a system of equations. Štefan Gyürki A few families of non-schurian association schemes 38/ 40

39 Preliminaries Faradžev I.A., Klin M.H.:Computer package for computations with coherent configurations, Proc. ISSAC-91, pp Groups, algorithms and programming (GAP), Hafner P.R.: Geometric realization of the graphs of McKay-Miller-Širáň, J. Comb. Th. B, 90(2) (2004), Klin M.H., Muzychuk M.E., Pech C., Woldar A.J., Zieschang P-H.: Association schemes on 28 points as mergings of a half-homogeneous coherent configuration, Eur. J. Combin. 28(7) (2007), Wild P.: Biaffine planes and divisible semiplanes, J. Geom., 25(2) (1985), Štefan Gyürki A few families of non-schurian association schemes 39/ 40

40 Thank you Preliminaries Thank you for your attention. Štefan Gyürki A few families of non-schurian association schemes 40/ 40